To date, the Hubbard model and its strong coupling limit, the t-J model, serve as the canonical model for strongly correlated electron systems in solids. Approximating the Coulomb interaction by only the on-site term (Hubbard U-term), however, may not be sufficient to describe the essential physics of interacting electron systems. We develop a more complete model in which all the next leading order terms besides the on-site term are retained. Moreover, we discuss how the inclusion of these neglected interaction terms in the Hubbard model changes the t-J model.
To date, the Hubbard model and its strong coupling limit, the t-J model, serve as the canonical model for strongly correlated electron systems in solids. Approximating the Coulomb interaction by only the on-site term (Hubbard U-term), however, may not be sufficient to describe the essential physics of interacting electron systems. We develop a more complete model in which all the next leading order terms besides the on-site term are retained. Moreover, we discuss how the inclusion of these neglected interaction terms in the Hubbard model changes the t-J model.
We develop finite-temperature strong-coupling expansions for the
SU(N) Hubbard model in powers of $\ensuremath{\beta}t, w=exp(\ensuremath{-}\ensuremath{\beta}U)$ and $\frac{1}{\ensuremath{\beta}U}$ for arbitrary filling. The expansions are done in the grand canonical ensemble and are most useful at a density of one particle per site, where for
U larger than or of order of the bandwidth, the expansions converge over a wide temperature range ${t}^{2}/U\phantom{\rule{4pt}{0ex}}\ensuremath{\lesssim}\phantom{\rule{4pt}{0ex}}T\phantom{\rule{4pt}{0ex}}\ensuremath{\lesssim}\phantom{\rule{4pt}{0ex}}10U$. By taking the limit $w\ensuremath{\rightarrow}0$, valid at temperatures much less than
U, the expansions turn into a high-temperature expansion for a dressed
SU(N) Heisenberg model that includes nearest-neighbor exchange, further neighbor exchanges, and ring exchanges known from the
T=0 perturbation theory of the SU(2) Hubbard model. Below a filling of one particle per site, the $w\ensuremath{\rightarrow}0$ limit corresponds to an effective $t\text{\ensuremath{-}}J$ model. The onset of strong correlations can be identified by a plateaulike behavior in the entropy as a function of temperature. At small deviations from one particle per site, the expansions can be arranged in powers of a small parameter $\ensuremath{\delta}=1\ensuremath{-}n$, the deviation from one particle per site, where the leading $\ensuremath{\beta}t$-dependent terms correspond to holes sloshing around in a disordered
SU(N) background. We use these expansions to calculate the thermodynamic properties of the model at moderate and high temperatures over a wide parameter range.
We study the effect of the correlated hopping term and the intersite Coulomb interaction term on principal features of the
d-wave superconducting (SC) state, in both the electron- and hole-doped regimes within the $t\text{\ensuremath{-}}J\text{\ensuremath{-}}U$ model. In our analysis, we use the approach based on the diagrammatic expansion of the Gutzwiller wave function (DE-GWF), which allows us to go beyond the renormalized mean-field theory (RMFT). We show that the correlated hopping term enhances the pairing at the electron-doped side of the phase diagram. Moreover, the so-called non-BCS regime (which manifests itself by the negative kinetic energy gain at the transition to the SC phase) is narrowed down with the increasing magnitude of the correlated hopping $\ensuremath{\sim}K$. Also, the doping dependencies of the nodal Fermi velocity and Fermi momentum, as well as the average number of double occupancies, are analyzed with reference to the experimental data for selected values of the parameter
K. For the sake of completeness, the influence of the intersite Coulomb repulsion on the obtained results is provided. Additionally, selected results concerning the Hubbard-model case are also presented. A complete model with all two-site interactions is briefly discussed in Appendix for reference.
The renormalization of the electron-phonon interaction by strong electronic correlations is studied using a one-band Hubbard model with infinite repulsion and nearest (t model) and nearest and second-nearest (tt\ensuremath{'} model) neighbor hopping terms and an on-site electron-phonon coupling. Using Hubbard's X operators and an extension from 2 to N degrees of freedom for the electrons the leading contributions for the electron self-energy and the vertex function in 1/N and the electron-phonon coupling constant are given and numerically evaluated for a square lattice. We find that the momentum dependence of the vertex function depends strongly on doping: For large dopings it is rather weak, with decreasing doping it becomes more and more pronounced leading to a strong reduction of the vertex at larger momentum transfers until, for very small dopings, the vertex essentially consists of a forward scattering peak with a width proportional to the doping. This behavior occurs both in the t and the tt\ensuremath{'} models and also in one dimension where analytic expressions are derived. Correlation effects also change ${\mathrm{\ensuremath{\alpha}}}^{2}$F in general: The full-symmetric component ${\mathrm{\ensuremath{\alpha}}}^{2}$
F1 is in the t model somewhat, in the tt\ensuremath{'} model rather strongly suppressed, especially near half-filling; the other symmetry components ${\mathrm{\ensuremath{\alpha}}}^{2}$
Fi with i=2,...,5 increase strongly with decreasing doping and are of similar magnitude as ${\mathrm{\ensuremath{\alpha}}}^{2}$
F1 near half-filling.Including also direct Coulomb repulsion nontrivial symmetries such as d wave become more stable than the s-wave order parameter below a critical value for the doping even for the considered phonon-mediated superconductivity. Most dramatic, however, is the quenching of the resistivity due to electron-phonon scattering both in the t and the tt\ensuremath{'} models at intermediate and small dopings. This result may explain the absence of phonon features in the experimental transport coefficients of high-
Tc compounds. \textcopyright{} 1996 The American Physical Society.
The Hubbard model with strong correlation is examined. The mean-field approximation (MFA) based on the best single-site approximation (SSA) is rigorous in the limit of high dimension of d =+∞. The best SSA is reduced to solving the Anderson model. The ground state is a heavy-electron liquid, presumably superconducting, in the weak-coupling regime, which is approximately defined by g ≈ J ( Q )/4 T K <1 with J ( Q ) being the maximum value of exchange interactions and T K the Kondo temperature to show the energy scale of heavy electrons. In the strong-coupling regime defined by g >1, on the other hand, the ground state is a MFA-type antiferromagnetic state.
It is important to understand the mechanism of high-temperature superconductivity. It is obvious that the interaction with large energy scale is responsible for high critical temperature T c . The Coulomb interaction is one of candidates that bring about high-temperature superconductivity because its characteristic energy is of the order of eV. There have been many works for the Hubbard model including three-band d-p model with the on-site Coulomb repulsion to investigate a possibility of high-temperature superconductivity. It is, of course, not trivial whether the on-site Coulomb interaction leads to a pairing interaction between two electrons. We argue that high-temperature superconductivity is possible in the strongly correlated region by using the variational Monte Carlo method for the two-dimensional t-U-J-V model. The exchange interaction J and the nearest-neighbour attractive interaction V cooperate with U and will act to enhance the critical temperature.